Download Effects of pulsatile blood flow in large vessels on thermal dose

Effects of pulsatile blood flow in large vessels on thermal dose distribution
during thermal therapy
Tzyy-Leng Horng
Department of Applied Mathematics, Feng Chia University, Taichung, Taiwan
Win-Li Lin
Institute of Biomedical Engineering, National Taiwan University, Taipei, Taiwan and Medical Engineering
Research Division, National Health Research Institutes, Miaoli, Taiwan
Chihng-Tsung Liauh
Department of Mechanical Engineering, Kun Shan University of Technology, Tainan, Taiwan
Tzu-Ching Shiha兲
Department of Medical Radiology Technology, China Medical University and Department of Radiology,
China Medical University Hospital, Taichung, Taiwan
共Received 1 November 2006; revised 3 February 2007; accepted for publication 6 February 2007;
published 16 March 2007兲
The aim of this study is to evaluate the effect of pulsatile blood flow in thermally significant blood
vessels on the thermal lesion region during thermal therapy of tumor. A sinusoidally pulsatile
velocity profile for blood flow was employed to simulate the cyclic effect of the heart beat on the
blood flow. The evolution of temperature field was governed by the energy transport equation for
blood flow together with Pennes’ bioheat equation for perfused tissue encircling the blood vessel.
The governing equations were numerically solved by a novel multi-block Chebyshev pseudospectral method and the accumulated thermal dose in tissue was computed. Numerical results show that
pulsatile velocity profile, with various combinations of pulsatile amplitude and frequency, has little
difference in effect on the thermal lesion region of tissue compared with uniform or parabolic
velocity profile. However, some minor differences on the thermal lesion region of blood vessel is
observed for middle-sized blood vessel. This consequence suggests that, in this kind of problem, we
may as well do the simulation simply by a steady uniform velocity profile for blood flow. © 2007
American Association of Physicists in Medicine. 关DOI: 10.1118/1.2712415兴
Key words: pulsatile blood flow, bioheat transfer, thermal dose, Chebyshev pseudospectral method
I. INTRODUCTION
Blood flow can remarkably affect the temperature distributions in tumor tissues during thermal therapies, especially
with the presence of large blood vessels.1–6 Thermally significant blood vessels 共larger than 0.2 mm in diameter兲 cool
tissue during thermal treatments, making it very difficult to
cover the whole tumor volume with therapeutic thermal
exposure.7–9 Shih et al.9 found that the short-duration highintensity heating can treat a target tumor inside with a blood
vessel diameter of 0.2 mm when using the total power energy density up to 120 J cm−3. Moreover, Becker and
Kuznetsov7 found that the target region tissue was found to
have a lower thermal damage in the presence of a blood
vessel, and this cooling effect is more pronounced as the
blood vessel gets larger 共larger than 0.6 mm in diameter兲.
These phenomena motivated us to thoroughly study the effects of thermally significant blood vessels on thermal
therapy of tumor.
The cyclic nature of the heart pump generates pulsatile
blood flow in all arteries.10–20 This periodic-in-time velocity
profile for pulsatile blood flow, driven by an oscillating pressure gradient, inside a circular blood vessel was first studied
by Womersley.20 Loudon and Tordesillas21 used dimensionless Womersley number to characterize the pulsatile fre1312
Med. Phys. 34 „4…, April 2007
quency of blood flow. They showed that when Womersley
number is small, the flow tracks the oscillating pressure gradient faithfully and the velocity profile exhibits a parabolic
shape. Nevertheless, when Womersley number is large, the
phase difference between velocity and pressure gradient becomes larger and the velocity profile exhibits a shape of two
peaks.
Generally in hyperthermia treatment of tumors, blood vessels with a diameter larger than 0.5 mm can no longer be
included in blood perfusion and must be treated
individually22–26 because temperature difference between
blood and surrounding tissue is significant. Heat transfer in
tissue within the region near a large blood vessel is especially influenced significantly by the flowing blood.27,28 Furthermore, Mohammed and Verhey28 utilized a finite element
method to simulate laser interstitial thermo therapy on anatomical inhomogeneous regions by considering blood perfusion only and also performed in in vitro experiment on porcine muscle tissue. Their numerical results had 5% to 20%
discrepancy in thermal lesion compared with the experiment.
This indicates the importance of an appropriate thermal
model to achieve reliable results from numerical simulations.
So far no numerical result considering the coupled heat
transfer problem of both a large blood vessel with pulsatile
0094-2405/2007/34„4…/1312/9/$23.00
© 2007 Am. Assoc. Phys. Med.
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Horng et al.: Pulsatile blood flow
blood flow inside and its surrounding tissue is ever reported,
and the whole physics remains poorly understood. Craciunescu and Clegg15 numerically solved the axis-symmetric
Navier-Stokes equations to obtain the pulsatile velocity profile and then computed the energy transport equation to obtain the temperature distribution of pulsating blood flow in a
blood vessel with sizes of aorta, large arteries, terminal arterial branches, and arterioles. Some cases of their simulation
have pulsation of large amplitude such that reversal of blood
flow occurs in large vessels 共i.e., 3 mm in diameter and
larger兲. However, they only focused on the temperature distribution within a blood vessel and did not consider that for
its surrounding tissue. Shih et al.8 computed the coupled heat
transfer problem of both a large blood vessel and its surrounding tissue. However, they only assumed the velocity
profile of the blood flow inside the vessel to be simply steady
and uniform. They found that the thermal dose would be
insufficient for treatment in the region near blood vessel
when the radius and flow rate are large. Shih et al.9 extended
Shih et al.8 to investigate the influence of various heating
speeds with the total heat applied held equal and found the
thermal dose deficit in the region near blood vessel is less
severe when the heating speed is large. Here we studied this
coupled heat transfer problem between a thermally significant blood vessel and its surrounding tissue during heat treatment by considering the velocity profile of the blood flow
being uniform, parabolic, and pulsatile under various radii of
vessel, pulsating frequencies and amplitudes 共in the case of
pulsatile velocity profile兲, and heating speeds. The main object is to realize the effects of flow velocity profile, size of
vessel, flow arte, and heating speed on thermal dose distribution of the surrounding tissue during heat treatment.
Though many factors about blood vessels have been brought
into consideration in the current study compared with existing literatures, it is still far from being complete. Many other
factors such as vessel wall elasticity, porosity, rheology of
blood, realistic shape of blood pressure wave, possible
counter current flow, and even turbulence in pulsatile blood
flow are either ignored or simplified in the current
study.11,12,29 Here the whole paper is organized as follows:
the mathematical model and its governing equations followed by its numerical method are described in the section
of mathematical model and numerical method. The numerical results are presented in the section of results and discussions with the effects of flow velocity profile, size of vessel,
flow rate, and heating speed on thermal dose distribution
discussed. The final section is conclusion to summarize all
key physics in this study.
II. MATHEMATICAL MODEL AND NUMERICAL
METHOD
A. Velocity profile of pulsatile blood flow in a circular
blood vessel
Besides being simply steady uniform or parabolic, the
pulsation of blood flow is considered here and its velocity
profile is derived as follows. Here, we assume the blood flow
to be incompressible, laminar and Newtonian obeying
Medical Physics, Vol. 34, No. 4, April 2007
1313
FIG. 1. Velocity profile of pulsatile blood flow at different phases for r0
= 0.5 mm, f = 1 Hz, ␣ = 0.64213, fac= 0.2. 共a兲 oscillating part of total velocity, 共b兲 total velocity.
Navier-Stokes equations in a cylindrical impermeable blood
vessel with rigid wall of radius r0. The axial HagenPoiseuille steady parabolic velocity profile is12
w=−
1 2 2 dp
共r − r 兲 .
4␮ 0
dz
共1兲
For pulsatile flows, the pressure gradient above is no more a
constant, and is in fact periodic in time due to the rhythmic
nature of beating heart. To mimic the realistic shape of pressure gradient in time, it requests at least 11 harmonics,29 and
here we simply consider the fundamental one only for the
feasibility of analysis. With this additional sinusoidal component in time, the corresponding pressure gradient, without
loss of major physics, along the z-axis of blood vessel can be
assumed periodically as
⳵p
= c0 + c1 sin ␻t
⳵z
= c0 + c1
ei␻t − e−i␻t
2i
= c0 + ĉ1ei␻t + ĉ−1e−i␻t ,
共2兲
where ␻ = 2␲ f is the angular frequency; the period is denoted
as T = 1 / f; ĉ1 = c1 / 2i, ĉ−1 = −c1 / 2i. The corresponding axial
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Horng et al.: Pulsatile blood flow
1314
velocity profile W共r , t兲 can be similarly expressed in the form
as
W = w + w1ei␻t + w−1e−i␻t ,
w̄ =
共3兲
冉 冊
by Eq. 共1兲. The volume flow rate
with
averaged over the period T is
Q̇avg =
2␲
T
冕冕
0
r0
Wrdrdt = −
0
␲r40
8␮
W = 2w̄ 1 −
共4兲
c0 ,
冉 冊
r2
+ w1ei␻t + w−1e−i␻t
r20
冤
冤
冉
冉
冊
冊
冥
冉 冊
冤
冉 冊冥
冉 冊
冦冤 冉 冊冥 冧
冉 冊
r
J0 ␣ i3/2
r0
iĉ1
+
1−
␳␻
r
J0 ␣ i3/2
r0
冉 冊
r
J0 ␣ i3/2
r0
c1
+
1−
r
2␳␻
J0 ␣ i3/2
r0
2
r
= 2w̄ 1 − 2
r0
2
r
= 2w̄ 1 − 2
r0
冉 冊
r2
c1
= 2w̄ 1 − 2 +
Re
␳␻
r0
冤
r0
denotes the Womersley number; ␳ and ␯ the density and
kinematic viscosity of blood, respectively; Re共·兲 the real
part; J0 is the Bessel function of the first kind of order zero.
Also, we denote here
冒冉 冊
−
8␮w̄
r20
to characterize the relative intensity of pulsation in blood
flow. In current study, the value of fac ranges from 0.2 to 1,
that is not too large to cause reverse flow judging from possible practical situations. The Womersley number ␣ is used
to describe the competition between transient inertia force
and viscous force. If the Womersley number ␣ is large, the
effect of viscosity cannot propagate far from the wall, and
the flow in the central portion of tube acts like inviscid flow
and is chiefly determined by the balance of inertia force and
pressure gradient. On the other hand, when the Womersley
number ␣ is large, the velocity profile in a pulsatile flow is
Medical Physics, Vol. 34, No. 4, April 2007
冥
共6兲
e−i␻t
冊
冊
冥
e−i␻t
e i␻t ,
冑␯/␻
fac = c1/c0 = c1
冉
冉
冊
r
J0 ␣ i3/2
r0
c1
e i␻t +
1−
r
2␳␻
J0 ␣ i3/2
r0
where
␣=
冉
r
J0 ␣ i5/2
r0
iĉ−1
e i␻t −
1−
␳␻
J0共␣i5/2兲
r
J0 ␣ i3/2
r0
1−
r
J0 ␣ i3/2
r0
r2
+ w1ei␻t + w−1e−i␻t .
r20
The sinusoidal components w1 and w−1 can be determined by
solving Navier-Stokes equations, and Eq. 共6兲 can be expressed as follows by Fung:12
and the average velocity in space and time is, therefore,
W = 2w̄ 1 −
共5兲
The velocity profile can then be further expressed as
w = −1 / 4␮共r20 − r2兲c0
T
c0r20
Q̇avg
=
−
.
8␮
␲r20
共7兲
expected to be relatively blunt compared with the steady
parabolic profile of Poiseuille flow. When Womersley number ␣ is larger, the pulsating velocity profile may even have
two peaks. However, this two-peak velocity profile did not
occur in the current study considering the vessel diameter
used here ranging from 0.2 to 2 mm and heart beat frequency 1 to 2 Hz that makes maximum Womersley number
1.8162 only. Figures 1共a兲 and 1共b兲 show the velocity profile
of oscillating component and the total, respectively, for various phases in a complete cycle for a typical case in the current study 共r0 = 0.5 mm, f = 1 Hz, ␣ = 0.642 13, fac= 0.2兲.
Neither two-peak behavior nor reverse flow is observed in
the velocity profile.
B. Governing equations
Here we assume that the absorbed power density in blood
and tissue is equal to the heating power density. The axissymmetric geometric configuration considered here is a cylindrical perfused tissue, including tumor and normal tissues,
with a coaxial rigid blood vessel inside and throughout the
tissue as shown in Fig. 2. The whole computational domain
is bounded by r = rmax, z = 0, and z = zmax; the blood vessel is
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Horng et al.: Pulsatile blood flow
1315
TABLE II. Three different heating schemes used in current simulations.
Case
Absorbed power density Q 共W cm−3兲
Heating duration th 共s兲
Total absorbed energy density 共J cm−3兲
FIG. 2. Geometric configuration in current simulations. 共a兲 The treatment
target 共heating target兲 is specified as z1 艋 z 艋 z2, 0 艋 r 艋 r1, with z1 = 5 mm,
z2 = 15 mm, r1 = 5 mm considered here. 共b兲 Schematic illustration of the
three kinds of velocity profile of blood flow in blood vessels. Left-steady
uniform velocity profile, middle-steady parabolic velocity profile, rightpulsatile velocity profile.
surrounded by r = r0, z = 0, and z = zmax; the heating target 共tumor and a part of blood vessel inside the tumor兲 is bounded
by r = r1, z = z1, and z = z2. The diameters of blood vessels and
their associated average flow velocities considered in the current study are presented in Table I.
The governing equations for the temperature evolution are
energy transport equations shown in Eqs. 共8兲 for tissue and
共9兲 for blood vessel:
␳ tc t
I
II
III
50
2
100
10
10
100
2
50
100
and specific heat, respectively, that are all assumed to be
constant; Q共r , z , t兲 is the power of heat added axissymmetrically; Wb is the perfusion mass flow rate; Ta is the
ambient temperature that is usually set to be 37 ° C; w共r , t兲 is
that axial velocity of blood flow; subscripts t and b represent
tissue and blood, respectively. Equation 共8兲 is in fact Pennes’
bioheat transfer equation30 in cylindrical coordinates for tissue with the heat sink −Wbcb共Tt − Ta兲 to describe the perfusion effect by the network of microvascular blood flow.
Equation 共9兲 is the conventional energy transport equation
for blood vessel with both convection and diffusion taken
into account.
The initial condition is
Tt共r,z,0兲 = Tb共r,z,0兲 = Ta = 37 ° C.
共10兲
At the interface conditions between the blood vessel and tissue are continuity of temperature and heat flux
Tt = Tb
kt
at ⌫,
⳵Tt
⳵Tb
= kb
⳵n
⳵n
共11兲
at ⌫,
共12兲
冋 冉 冊 册
1 ⳵ ⳵Tt
⳵Tt
⳵ 2T t
= kt
r
+ 2 − Wbcb共Tt − Ta兲
⳵t
r ⳵r ⳵r
⳵z
+ Qt共r,z,t兲,
␳ bc b
冉
冊 冋 冉 冊
册
共8兲
1 ⳵ ⳵Tb
⳵Tb
⳵Tb
⳵ 2T b
+w
= kb
r
+ 2 + Qb共r,z,t兲,
⳵z
⳵t
r ⳵r ⳵r
⳵z
共9兲
where T共r , z , t兲 denotes the temperature that is distributed
axis-symmetrically; ␳ , k , c are density, thermal conductivity,
TABLE I. List of the blood vessel parameters used in current simulations
共Ref. 8兲.
Diameter 共mm兲
Average blood velocity in tumor 共w̄兲 共mm s−1兲
0.2
0.6
1.0
1.4
2.0
3.4
6
8
10.5
20
Medical Physics, Vol. 34, No. 4, April 2007
FIG. 3. The computational domain is decomposed into nine rectangular
blocks. Blocks 1, 4, and 7 are blood vessel; block 5 is the tumor tissue; the
other blocks are normal tissue. Blocks 4 and 5 are heating zone. Here the
mesh resolution is 10⫻ 10 in each block. Due to the feature of Chebyshev
collocation points, grids are clustered more densely near boundary/interface
in each block. In this figure rmax = 10 mm, zmax = 100 mm, r0 = 2 mm, r1
= 5 mm, z1 = 10 mm, z2 = 20 mm.
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Horng et al.: Pulsatile blood flow
1316
FIG. 5. Effect of the frequency of the pulsatile blood flow on the thermal
lesion region 共shaded region兲 for the blood vessels 共a兲, 共f兲, 共k兲 0.2 mm; 共b兲,
共g兲, 共l兲 0.6 mm; 共c兲, 共h兲, 共m兲 1 mm; 共d兲, 共i兲, 共n兲 1.4 mm; 共e兲, 共j兲, 共o兲 2 mm in
diameter. 共a兲–共e兲 with the frequency 1 Hz, 共f兲–共j兲 with the frequency 1.5 Hz,
and 共k兲–共o兲 with the frequency 2 Hz. The blood vessel boundaries are denoted with the horizontal dashed lines. The heated target region is denoted
by a square with dashed lines. Here r1 = 5 mm, Wb = 2 kg m−3 s−1, ␳b = ␳t
= 1050 kg m−3, cb = ct = 3770 J kg−1 ° C−1, kb = kt = 0.5 W m−3 ° C−1. The target is heated in the way of the heating case I 共i.e., Qt = Qb = 50 W cm−3, and
the heating duration= 2 s兲 in Table II.
FIG. 4. Effect of the steady velocity profiles of blood flow on the thermal
lesion region 共shaded region兲 for the blood vessels: 共a兲, 共f兲 0.2 mm; 共b兲, 共g兲
0.6 mm; 共c兲, 共h兲 1 mm; 共d兲, 共i兲 1.4 mm; 共e兲, 共j兲 2 mm in diameter. 共a兲–共e兲 are
results of a uniform velocity profile, and 共f兲–共j兲 are results of a parabolic
one. The blood vessel boundaries are denoted with the horizontal dashed
lines. The heated target region 共tumor兲 is denoted by a square with dashed
lines. Here r1 = 5 mm, Wb = 2 kg m−3 s−1, ␳b = ␳t = 1050 kg m−3, cb = ct
= 3770 J kg−1 ° C−1, kb = kt = 0.5 W m−3 ° C−1. The target is heated in the way
of the heating case I 共i.e., Qt = Qb = 50 W cm−3, and the heating duration
= 2 s兲 in Table II.
where ⌫ denotes the interface between blood vessel and tissue, and n denotes the direction normal to ⌫. At r = 0, pole
condition is applied for blood vessel by axis-symmetry,
Medical Physics, Vol. 34, No. 4, April 2007
⳵Tb
= 0.
⳵r
共13兲
Finally, the boundary condition at r = rmax, z = 0, and z = zmax
is temperature all equal to the ambient one,
Tt 共or Tb兲 = Ta = 37 ° C,
共14兲
except the blood part at z = zmax using convective boundary
condition,
⳵Tb
⳵Tb
= 0,
+w
⳵z
⳵t
at z = zmax .
共15兲
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1317
FIG. 6. Comparison of the effect of various velocity
profiles on the thermal lesion region for 1-mm-diam
blood vessel with 共a兲 steady uniform velocity profile,
共b兲 steady parabolic velocity profile, 共c兲 1 Hz and fac
= 0.2 pulsatile velocity profile, 共d兲 1.5 Hz and fac= 0.2
pulsatile velocity profile, 共e兲 2 Hz and fac= 0.2 pulsatile velocity profile.
C. Calculation of thermal dose
The accumulated thermal dose to tissue is a function of
heating duration and the temperature level. The estimate of
tissue damage is based on the thermal dose of which the
formula was proposed by Sapareto and Dewey.31 The thermal dose or equivalent minutes at 43 ° C 共EM43兲 is shown as
follows:
EM43共in min . 兲 =
冕
tf
RT−43dt,
共16兲
0
where R = 2 for T 艌 43 ° C, R = 4 for 37 ° C ⬍ T ⬍ 43 ° C, and
t f = 90 s or 120 s in current simulation. The threshold dose
for necrosis is EM43 = 240 min for muscle tissue.32 Therefore,
the region encircled by the level curve EM43 = 240 min is
taken as the thermal lesion region in this study. Covering
tumor tissue but not normal tissue by thermal lesion region
as full as possible is most desired in the thermal treatment,
Medical Physics, Vol. 34, No. 4, April 2007
and the deficit would serve as an evaluation of the effectiveness of the treatment. This region is greatly influenced by
size of blood vessel31 and the heating speed.32 Here three
different heating schemes characterizing different heating
speeds under the same amount of heat added 共100 J cm−3
from preliminary energy analysis in lump33兲 are depicted in
Table II.
D. Numerical method
Here we used method of lines 共MOL兲 to solve Eqs.
共8兲–共15兲. First, we employed multi-block Chebyshev pseudospectral method to discretize governing equations 共8兲 and
共9兲, boundary and interface conditions 共10兲–共15兲 in space
into a semi-discrete system in time. This coupled system
consists of ordinary differential equations in time, mainly
from Eqs. 共8兲 and 共9兲, and algebraic equations, chiefly from
boundary and interface conditions 共10兲–共15兲. We can then
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FIG. 9. Effect of fac of the pulsatile blood flow on the thermal lesion region
with the pulsatile frequency being 1 Hz. The target is heated in the way of
the heating case I 共i.e., Qt = Qb = 50 W cm−3, and the heating duration= 2 s兲
in Table II. 共a兲 fac= 0.2, 共b兲 fac= 0.5, 共c兲 fac= 0.8, and 共d兲 fac= 1.
FIG. 7. Comparison of the effect of different heating schemes and blood
vessel diameters on the thermal lesion region with steady uniform velocity
profile. The solid and dashed lines represent the heating cases I and II, and
the shaded area represents the heating case III. 共a兲 The diameter of the blood
vessel is 0.2 mm. 共b兲 The diameter of the blood vessel is 1 mm.
solve this system of differential-algebraic equations 共DAE’s兲
by many efficient DAE solvers available from most numerical libraries, and here we used popular MATLAB index-1
DAE solver ode15s in our computation. Multi-block Chebyshev pseudospectral method first decomposes the computational domain into several blocks, based on heterogeneity of
thermal properties or heating, and discretizes equations in
space in each block by conventional Chebyshev pseudospectral method. Details about Chebyshev pseudospectral meth-
FIG. 8. The temperature contours for the case of blood vessel diameter
1 mm with steady uniform blood flow under case-I heating 共i.e., Qt = Qb
= 50 W cm−3, and the heating duration= 2 s兲 at the time of power off 共t
= 2 s兲 with the highest temperature reaching 62.108 ° C.
Medical Physics, Vol. 34, No. 4, April 2007
ods are referred to Canuto et al.,34 Trefethen,35 and
Fornberg.36 Here the computational domain is decomposed
to nine rectangular blocks to cope with blood vessel and
heating zone as shown in Fig. 3. We can observe from Fig. 3
that the grids are clustered more densely near boundary/
FIG. 10. Effect of fac of the pulsatile blood flow on the thermal lesion
region with the pulsatile frequency being 1.5 Hz. The target is heated in the
way of the heating case I 共i.e., Qt = Qb = 50 W cm−3, and the heating
duration= 2 s兲 in Table II. 共a兲 fac= 0.2, 共b兲 fac= 0.5, 共c兲 fac= 0.8, and 共d兲
fac= 1.
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Horng et al.: Pulsatile blood flow
1319
interface due to the characteristics of Chebyshev collocation
points.34–36 This feature makes the computational accuracy
and efficiency far better than the one without any decomposition. In short, owing to the intrinsic high order of accuracy
of pseudospectral methods, grid size is not that restricted as
finite difference/element methods and, hence, the overall
computation is very efficient.
III. RESULTS AND DISCUSSIONS
Figure 4 compares the effects of steady uniform and parabolic velocity profiles for blood flow on thermal lesion region with heating scheme I in Table II, which has the least
thermal dose deficit as described in Shih et al.9 Likewise,
Fig. 5 compares the effects of pulsatile blood flow with various pulsatile frequencies 共1, 1.5, and 2 Hz兲 with the relative
intensity of pulsation fac= 0.2 on thermal lesion region with
heating Scheme I in Table II. Figures 4 and 5 generally show
that there is almost no difference in thermal lesion region
among all these velocity profiles under the same size of
blood vessel. Only minor difference of thermal lesion region
in blood vessel is observed in middle-sized blood vessels
共with diameter 0.6 and 1 mm兲. This minor difference is most
pronounced for blood vessel of diameter 1 mm and is particularly shown in Fig. 6. It shows that the length of thermal
lesion region in blood vessel is longest for uniform velocity
profile, shortest for parabolic and 1.5 Hz pulsatile ones, and
in between for 1 and 2 Hz pulsatile flows.
While the thermal lesion region is rather insensitive to the
velocity profile of blood flow, it is deeply influenced by the
size of blood vessel since the heat convection by the blood
flow in a blood vessel usually serves as a stronger heat sink
than the blood perfusion in tissue. That means temperature
would drop faster in a blood vessel than its surrounding tissue. This may cause deficit of thermal lesion region in blood
vessel and tissue nearby, which can be easily observed from
Figs. 4 and 5. Generally the deficit of thermal lesion region is
less for smaller vessels. In the case of smallest vessel here
共with diameter 0.2 mm兲, thermal lesion region almost covers
the whole blood vessel inside the tumor and the deficit is
naturally the least. For middle-sized vessels here 共with diameter 0.6 and 1 mm兲, the thermal lesion region in blood vessel
becomes smaller and shifted downstream. For large vessels
here 共with diameter 1.4 and 2 mm兲, there is a total deficit of
thermal lesion region in blood vessel, and this would cause
deficit in tumor tissue right near the blood vessel especially
at the upstream.
Besides diameters of blood vessel, the thermal lesion region is also very sensitive to the heating speed. Figure 7
generally shows larger thermal lesion region for faster heating, and the heating speed affects more pronouncedly when
the blood vessel is larger. As shown in Fig. 7共b兲, there exists
an obvious shift of thermal lesion region to the downstream
of blood vessel for the blood vessel with diameter 1 mm
when heating is fast, and this may cause unwanted thermal
injury in normal tissue nearby. Figure 8 particularly shows
the temperature contours for the case of blood vessel diameter of 1 mm under heating Scheme I in Fig. 7 at the time of
Medical Physics, Vol. 34, No. 4, April 2007
FIG. 11. Effect of fac of the pulsatile blood flow on the thermal lesion
region with the pulsatile frequency being 2 Hz. The target is heated in the
way of the heating case I 共i.e., Qt = Qb = 50 W cm−3, and the heating
duration= 2 s兲 in Table II. 共a兲 fac= 0.2, 共b兲 fac= 0.5, 共c兲 fac= 0.8, and 共d兲
fac= 1.
power off 共t = 2 s兲. Notice that the highest temperature occurs
right at the downstream of blood vessel. This causes the
thermal damage of the normal tissue nearby.
The effect of pulsation amplitude, in terms of relative intensity fac, of pulsatile blood flow on the thermal lesion
region generally has little difference among various fac’s,
except minor difference for middle-sized blood vessels. Figures 9–11 depicts this effect for a middle-sized blood vessel
of diameter 1 mm with fac= 0.2, 0.5, 0.8, and 1 and frequency being at 1 Hz 共Fig. 9兲, 1.5 Hz 共Fig. 10兲, and 2 Hz
共Fig. 11兲, respectively. When fac increases, the blood flows
more in a stick-slip fashion, and this may considerably influence the heat convection when incorporated with pulsation
frequency. Figures 9, 10, 11共c兲, and 11共d兲, with fac= 0.8 and
1, respectively, show obviously two-peak behavior in thermal dose contour at the downstream of blood vessel, which
is chiefly because of large pulsation amplitude.
IV. CONCLUSION
The current investigation shows that the effect of velocity
profile of blood flow, ranging from uniform, parabolic, to
pusatile ones, has almost no difference in thermal lesion region on tumor region and minor difference only on blood
vessel when the blood vessel is of middle size. This result
suggests that we might just as well use the simplest steady
uniform velocity profile to do the simulation without significant difference in thermal lesion region of tumor. In fact, the
thermal lesion region is much more sensitive to heating
speed and size of blood vessel. Some studies8,9,26,33 show
that faster heating would form a much better thermal lesion
1320
Horng et al.: Pulsatile blood flow
region, and it works best on small blood vessels with a better
covering of both the tumor and blood vessel by the thermal
lesion region since the heat convection by the blood flow in
the blood vessel is least. For large vessels, it has a total
deficit in blood vessel and some deficit in tumor near the
upstream of blood vessel. As to middle-sized vessels, a shift
of partially deficient thermal lesion region to the downstream
of blood vessel may cause unwanted thermal injury to the
normal tissue nearby. Furthermore, for middle-sized blood
vessels, pulsatile blood flow of large pulsation amplitudes
may further cause a two-peak behavior in the thermal dose
contour at the downstream of blood vessel.
ACKNOWLEDGMENT
This research was supported in part by the National Science Council of the Republic of China under Grant No.
NSC-94-2213-E-039-004.
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
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